Fig. 3: Kinematic parameter optimisation via gradient-descent minimisation.

a Processing of a target event: The target event, along with its reconstructed muon kinematics, is input to the transformer (depicted in blue). The transformer generates a set of possible kinematic combinations for all particles within the target event. These kinematics are subsequently forwarded to the gradient-descent minimiser (depicted in red), which leverages the generative adversarial network (GAN) to refine the kinematics and improve the correspondence with the target event. The diagram visually represents the different decompositions resulting from this process. b The plot shows two scenarios in a likelihood space (pre-computed for the kinetic energy of two most energetic protons of an arbitrary event: KE\({}_{{{{{{{{{\rm{proton}}}}}}}}}_{1}}\) and KE\({}_{{{{{{{{{\rm{proton}}}}}}}}}_{2}}\)). One starts from the transformer output and successfully reaches the target values, while the other begins at a random parameter space point and gets stuck in a local minimum. c Profiled negative log-likelihood \({{{{{{{\mathcal{L}}}}}}}}\) for the kinetic energy of the most energetic proton (KE\({}_{{{{{{{{{\rm{proton}}}}}}}}}_{1}}\)) of an arbitrary event, and the curve shows the 68% confidence interval determined by a \(\Delta {{{{{{{\mathcal{L}}}}}}}}\) of 1 for one degree of freedom. d The resolution of kinetic energy (KE), as determined through an analysis of sets of random and hard events (i.e., events where the image reconstructed from the transformer exceeded a predefined mean-squared-error threshold in comparison to the target image), was assessed for three distinct methodologies: the transformer and two gradient-descent techniques ("GAN (gr. descent 1)” and “GAN (gr. descent)'', as per Algorithms 1 and 2, respectively, from the “Gradient-descent minimisation of the image parameters” subsection of the “Methods” section). It illustrates the effectiveness of GAN-based minimisation in refining the kinematic parameters.